Strongly self-absorbing $C^{*}$-algebras

Toms, Andrew S.; Winter, Wilhelm

doi:10.1090/s0002-9947-07-04173-6

Cited by 239 publications

(322 citation statements)

References 37 publications

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“…We say that a second algebra A is Z-stable if A ⊗ Z ∼ = A. Z-stability is our third regularity property. It is very robust with respect to common constructions (see [61]). …”

Section: Regularity Propertiesmentioning

confidence: 99%

Regularity properties in the classification program for separable amenable C*-algebras

Elliott

Toms

2008

Bull. Amer. Math. Soc.

145

137

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Abstract. We report on recent progress in the program to classify separable amenable C * -algebras. Our emphasis is on the newly apparent role of regularity properties such as finite decomposition rank, strict comparison of positive elements, and Z-stability, and on the importance of the Cuntz semigroup. We include a brief history of the program's successes since 1989, a more detailed look at the Villadsen-type algebras which have so dramatically changed the landscape, and a collection of announcements on the structure and properties of the Cuntz semigroup.

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“…We say that a second algebra A is Z-stable if A ⊗ Z ∼ = A. Z-stability is our third regularity property. It is very robust with respect to common constructions (see [61]). …”

Section: Regularity Propertiesmentioning

confidence: 99%

Regularity properties in the classification program for separable amenable C*-algebras

Elliott

Toms

2008

Bull. Amer. Math. Soc.

145

137

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“…* -algebras by (1), and hence it is D-stable by [12,Thm. 4.3], since D is a separable unital strongly self-absorbing…”

Section: Proving the Main Result: The First Approachmentioning

confidence: 99%

“…However, at the present stage, it is not known whether the theorem also fails for infinite dimensional spaces X if D = O ∞ or D = Z. c of the unitary group of D is contained in U 0 (D), the connected component of 1. Indeed, the only role of K 1 -injectivity in the proofs from this paper as well as the proofs of the results from [12], [7] and [5] that we use here is to ensure that approximate unitary equivalence for any two unital * -homomorphisms D → D ⊗ A is induced by unitaries (u n ) n in U 0 (D). But as noted in [10] and [5] one can always choose the unitaries (u n ) n in U (D) c .…”

Section: Introductionmentioning

confidence: 99%

Trivialization of $\mathcal{C}(X)$-algebras with strongly self-absorbing fibres

Dǎdǎrlat¹,

Winter²

2008

Bul. Soc. Math. France

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Abstract. -Suppose A is a separable unital C(X)-algebra each fibre of which is isomorphic to the same strongly self-absorbing and K 1 -injective C * -algebra D. We show that A and C(X) ⊗ D are isomorphic as C(X)-algebras provided the compact Hausdorff space X is finite-dimensional. This statement is known not to extend to the infinite-dimensional case. Résumé (Trivialisation de C(X)-algèbres à fibres fortement auto-absorbantes)Soit A une C(X)-algèbre séparable unital dont chaque fibre est isomorphe à une même C * -algèbre D K 1 -injective et fortement auto-absorbante. Nous montrons que si l'espace compact et Hausdorff X est de dimension finie, alors A et C(X) ⊗ D sont isomorphes en tant que C(X)-algèbres. Ce resultat est connu pour ne pas s'étendre au cas des espaces de dimension infinie.

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“…Recall that a unital separable C * -algebra D (D = C) is called strongly self-absorbing if there is an isomorphism D ∼ → D⊗D that is approximately unitarily equivalent to the first factor embedding D → D⊗D sending d → d ⊗ 1 D [66]. Such C * -algebras turn out to be simple and nuclear.…”

Section: Strongly Self-absorbing Operadsmentioning

confidence: 99%

“…It is a noteworthy example of a strongly self-absorbing C * -algebra [66] with very interesting structural properties. Using a result from [13] (see also [31]) one can deduce that nonconnective algebraic K-theory agrees naturally with topological K-theory for O ∞ -stable C * -algebras.…”

Section: Introductionmentioning

confidence: 99%

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

Mahanta

2015

Math Phys Anal Geom

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Abstract. We develop an algebraic formalism for topological T-duality. More precisely, we show that topological T-duality actually induces an isomorphism between noncommutative motives that in turn implements the well-known isomorphism between twisted K-theories (up to a shift). In order to establish this result we model topological K-theory by algebraic K-theory. We also construct an E ∞ -operad starting from any strongly self-absorbing C * -algebra D. Then we show that there is a functorial topological K-theory symmetric spectrum construction K top Σ (−) on the category of separable C * -algebras, such that K top Σ (D) is an algebra over this operad; moreover, K top Σ (A⊗D) is a module over this algebra. Along the way we obtain a new symmetric spectra valued functorial model for the (connective) topological K-theory of C * -algebras. We also show that O ∞ -stable C * -algebras are K-regular providing evidence for a conjecture of Rosenberg. We conclude with an explicit description of the algebraic K-theory of ax + b-semigroup C * -algebras coming from number theory and that of O ∞ -stabilized noncommutative tori. IntroductionWithin the category of separable C * -algebras SC * the stable ones, i.e., those A ∈ SC * satisfying A⊗K ∼ = A, play a privileged role. For instance, it is known that they satisfy the Karoubi conjecture and appear very naturally in the context of twisted K-theory. The results in this article demonstrate that O ∞ -stable separable C * -algebras, i.e., those A ∈ SC * satisfying A⊗O ∞ ∼ = A, deserve a similar prominent status. Moreover, the Cuntz algebra O ∞ is strongly self-absorbing, which has several interesting ramifications.Ever since its inception by Kontsevich [33] the homological mirror symmetry conjecture has promoted rich interaction between geometry, algebra, and (higher) category theory. Mirror symmetry is related to T-duality via the Strominger-Yau-Zaslow conjecture [61] and hence we believe that it is worthwhile to have an algebraic formalism for T-duality at our disposal. One aspect of this theory is the Bunke-Schick topological T-duality, which has received a lot of attention in the mathematical literature. It is insensitive to subtle geometric structures but its mathematical underpinnings are very well understood [8,7,9]. One of the objectives of this article is to develop an algebraic formalism for topological T-duality relating it to the theory of noncommutative motives [34,35,65,45]. Along the way we obtain several interesting applications to algebraic K-theory and K-regularity of C * -algebras. The novelty 2010 Mathematics Subject Classification. 46L85, 19Dxx, 46L80, 57R56.

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Strongly self-absorbing $C^{*}$-algebras

Cited by 239 publications

References 37 publications

Regularity properties in the classification program for separable amenable C*-algebras

Regularity properties in the classification program for separable amenable C*-algebras

Trivialization of $\mathcal{C}(X)$-algebras with strongly self-absorbing fibres

Algebraic K-theory, K-regularity, and 𝕋 $\mathbb {T}$ -duality of 𝒪 ∞ $\mathcal {O}_{\infty }$ -stable C ∗-algebras

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